Remark 47.13.5. Let $A$ be a ring and let $I \subset A$ be an ideal. Set $B = A/I$. In this case the functor $\mathop{\mathrm{Hom}}\nolimits _ A(B, -)$ is equal to the functor

\[ \text{Mod}_ A \longrightarrow \text{Mod}_ B,\quad M \longmapsto M[I] \]

which sends $M$ to the submodule of $I$-torsion.

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